# Winter Semester 2018/2019

## Condition: The Geometry of Numerical Algorithms

**Lecturer:**Paul Breiding**Date:**Tuesday, 09:00 - 10:00**Room:**G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Condition numbers, complexity of numerical algorithms, loss of precision, average analysis, smoothed analysis**Literature:***Condition: the geometry of numerical algorithms*by Bürgisser and Cucker (Springer, 2013)

*The condition number of join decompositions*by Breiding and Vannieuwenhoven (SIAM J. Matrix Anal. and Appl., 39(1), 287–309)**Prerequisites:**Participants should have a good understanding of linear algebra and a basic knowledge about probability theory and differential geometry. Expertise in linear programming, algebraic geometry or multilinear algebra is helpful but not required.

## Abstract:

Numerical data rarely is given exact, but usually comes with errors due to noise, measurement errors or errors caused by prior calculations. Consequently, algorithms which take such data as input can't produce correct answers.

In this course we will learn about the concept of *condition numbers* and how it helps to understand how much the solution of a computational problem is changed, if the input data is perturbed. We will also discuss how condition numbers are related to the complexity of numerical algorithms. The concepts of average and smoothed analysis will be introduced.

We will consider the condition numbers of the following problems: linear equation solving, polynomial equation solving and computing tensor decompositions. If time permits, we will also cover condition numbers of problems in linear programming.

## Reading Seminar: Representation Theory - The Basics

**Lecturer:**Joscha Diehl**Date:**first lecture: Tuesday 16.10.2018 at 11:00h**Room:**Section F**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**representation theory; symmetric group; GL

## Abstract:

Groups are a fundamental construct to understand “symme- tries”. They are best understood by studying their action on linear vector spaces, that is via **linear representations **i.e. their concrete realization as matrices. This for example appears in

- harmonic analysis (think: the Fourier transform of periodic functions)
- tensor decomposition (a matrix (a 2-tensor) can be split into a symmetric part and an antisymmetric part; what about higher order tensors?)
- the study of certain Markov chains

This is a reading group of

*Fulton, William, and Joe Harris. Representation theory: a ﬁrst course. Vol. 129. Springer Science & Business Media, 2013. *

We will go through Chapter 1-15; at the speed required by the group. Starting essentially from scratch in Chapter 1, at the end we will have a solid understanding of representations of the symmetric group (permutations) and the general linear group (invertible matrices).

Each week one designated “leader” will guide the session, but all attendants are asked to read the relevant chapter before. We will work with examples (by hand and by code) as much as possible. We will meet for the ﬁrst time in the common area of Section F3, Tuesday 16.10.2018 at 11:00h.

## Introduction to Random Algebraic Geometry

**Lecturer:**Antonio Lerario**Date:**Tuesday, 09:00 - 10:00, from February 6 to March 27**Room:**G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Real Algebraic Geometry, Random Matrix Theory, Integral Geometry**Prerequisites:**Basic knowledge from Diﬀerential Geometry, Probability and Algebraic Geometry

## Abstract:

This course will deal with the basic problem of understanding the structure (e.g. the geometry and topology) of the set of solutions of real polynomial equations with *random* coeﬃcients. The simplest case of interest is the count of the number of real zeroes of a random univariate polynomial $$p(x) = a_{0} +· · ·+a_{d}x^{d}\: , where\: a_{0}, . . . , a_{d}$$ are random Gaussian variables – this problem is “classical” and was pioneered by Kac in the 40s. More generally algebraic geometers might be interested, for example, in the number of components of a* random* real plane curve of degree d, or in the *expected *number of real solutions of more advanced counting problems (e.g. enumerative problems). This is a very fresh and modern approach to real algebraic geometry: when the outcome is highly sensitive to the choice of the parameters (in general, there is no notion of “generic” in the real world), the attention is shifted to the “typical” situation. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with convex geometry, random matrix theory and random ﬁelds.

## Toric Varieties

**Lecturer:**Mateusz Michalek**Date:**Friday, 11:00 - 12:30**Room:**G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Algebraic Varieties, Lattice Polytopes, Complex Torus, Lattice Cones, Divisors, Orbits, Singularities**Prerequisites:**Basic knowledge of algebraic geometry and commutative algebra (say at the level of nonlinear algebra Ringvorlesung) or a lot of motivation and enough time to catch up

## Abstract:

Toric varieties form a special class of algebraic varieties, in some aspects generalizing aﬃne and projective spaces. Due to connections to combinatorics they also form one of the best understood classes. A lot of algebraic invariants turn out to behave in a particularly nice way for toric varieties. The aim of the lecture is to describe interactions of algebra, combinatorics and geometry; to learn about toric varieties, but at the same time gain experience in algebraic geometry.

We will be closely following the book ’Toric Varieties’ by Cox, Little and Schenck. The course will very actively involve participants - everyone will be expected to read the material in advance and some participants will be asked to present various parts. We hope to keep a fast pace, so that the course not only presents connections between polytopes/cones and toric varieties, but also describes more involved constructions including Cox rings, Picard and class groups, canonical divisor, sheaf co-homology, various types of singularities (including Gorenstein, Cohen-Macaulay, rational, normal) etc.

## Geometric and Algebraic Combinatorics

**Lecturer:**Bernd Sturmfels**Date:**October, 29 – November 2 and November 16**Room:**tba**Language:**English**Target audience:**PhD students**Prerequisites:**Mathematical maturity of an advanced graduate student. Interest in combinatorics.**Remarks:**For details check https://math.berkeley.edu/˜bernd/nov18.html

## Abstract:

The problem of the generation of random dynamical systems by stochastic partial diﬀerential equations (SPDE) is one of the open problems in the application of dynamical systems theory to SPDE. Despite its fundamental nature, most results are restricted to ”simple” random perturbations of aﬃne-linear structure. However, as we will see in this course, applications like scaling limits of particle systems with interaction and branching and non-equilibrium statistical mechanics, lead to porous media equations perturbed by nonlinear multiplicative or nonlinear conservative noise. We will ﬁrst convince ourselves that established methods such as the variational approach to SPDE cannot be applied to these equations, let alone prove the generation of random dynamical systems. Then, based on entropy and kinetic theory we will prove their well-posedness. This will lead to a strong notion of uniqueness, so-called path-by-path uniqueness, based on rough path theory which in turn proves the generation of a corresponding random dynamical system and opens the way to a qualitative analysis of the (stochastic) ﬂow of the solutions.

## Other Lectures at MPI MIS

Please follow this link for other regular lectures at the Max-Planck-Institute.